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Grothendieck's Galois theory
In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s.
The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted \( \hat{\Z} \), which is the inverse limit of the cyclic additive groups Z/nZ — or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group, so that it is specified by giving some permutation of X.
In the above example, a connection with classical Galois theory can be seen by regarding \( \hat{\Z} \) as the profinite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the zn map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n.Z of the fundamental group of the punctured disk.
The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor Φ, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type
- G ≅ Aut(Φ),
the latter being the group of automorphisms (self-natural equivalences) of Φ. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite.
To see how this applies to the case of fields, one has to study the tensor product of fields. Later developments in topos theory make this all part of a theory of atomic toposes.
References
Grothendieck, A. et al. (1971). SGA1 Revêtements étales et groupe fondamental, 1960–1961'. Lecture Notes in Mathematics 224. Springer Verlag.
Joyal, André; Tierney, Myles (1984). An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society. Proquest Info & Learning. ISBN 0-8218-2312-4.
Borceux, F. and Janelidze, G., Cambridge University Press (2001). Galois theories, ISBN 0-521-80309-8 (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)
Szamuely, T., Galois Groups and Fundamental Groups, Cambridge University Press, 2009.
Dubuc, E. J and de la Vega, C. S., On the Galois theory of Grothendieck, http://arxiv.org/abs/math/0009145v1
Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of GQ under the ℓ-adic cyclotomic character is \mathbf{Z}_\ell^\times. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of Qℓ with C they can be identified with bona fide Artin representations.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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